Minimal state variable solutions to Markov-switching rational expectations models

We develop a new method for deriving minimal state variable (MSV) equilibria of a general class of Markov switching rational expectations models and a new algorithm for computing these equilibria. We compare our approach to previously known algorithms, and we demonstrate that ours is both efficient and more reliable than previous methods in the sense that it is able to find MSV equilibria that previously known algorithms cannot. Further, our algorithm can find all possible MSV equilibria in models. This feature is essential if one is interested in using a likelihood based approach to estimation.     

Pooling forecasts in linear rational expectations models

Estimating linear rational expectations models in a limited-information setting requires replacing the expectations of future, endogenous variables either with instrumented, actual values or with forecast survey data. Applying the method of Gottfries and Persson [Empirical examinations of the information sets of economic agents. Quarterly Journal of Economics 103, 251–259], I show how to augment these methods with actual, future values of the endogenous variables to improve statistical efficiency. The method is illustrated with an application to the US hybrid new Keynesian Phillips curve, where traditional, lagged instruments and the median forecast from the Survey of Professional Forecasters both appear to miss significant information used by price-setters, so that forecast pooling with actual values improves the statistical fit to inflation.     

Solving generalized multivariate linear rational expectations models

We generalize the linear rational expectations solution method of Whiteman (1983) to the multivariate case. This facilitates the use of a generic exogenous driving process that must only satisfy covariance stationarity. Multivariate cross-equation restrictions linking the Wold representation of the exogenous process to the endogenous variables of the rational expectations model are obtained. We argue that this approach offers important insights into rational expectations models. We give two examples in the paper—an asset pricing model with incomplete information and a monetary model with observationally equivalent monetary-fiscal policy interactions. We relate our solution methodology to other popular approaches to solving multivariate linear rational expectations models, and provide user-friendly code that executes our approach.     

On the solutions in rational expectations models

This paper deals with the solutions to macroeconomic models with rational expectations. A first purpose is to demonstrate that two forms of the efficient-market equation commonly used in the literature may be treated as approximations of the same general equation, of different orders with respect to perturbative forces of the market. Second aim is to show how the perturbative terms can be given explicit forms. Finally, the paper addresses the question of finding contributions which can concur to explain the phenomenon usually referred to asstock price volatility.

---

Riassunto Si è interessati alla soluzione di modelli macroeconomici con aspettativa razionale. In primo luogo si vuole dimostrate come due forme differenti dell'equazione del mercato efficiente utilizzate nella letteratura possano essere considerate approssimazioni della stessa equazione generale, di ordini diversi rispetto alle forze perturbative del mercato. Si vuole quindi individuare una tecnica per dare forma esplicita ai termini perturbativi. Infine, si va alla ricerca di contributi che possano concorrere a spiegare il fenomeno noto comevolatilità dei prezzi del mercato azionario.

     

The design of feedback rules in linear stochastic rational expectations models

This paper is concerned with optimal and sub-optimal feedback rules for linear stochastic continuous time models with rational expectations. We consider four types of feedback rules: (1) the optimal but time-inconsistent rule which is available if the controller is able to commit himself or herself; (2) quasi-optimal and time-inconsistent rules of the form w = Dy where w is the vector of instruments, y the state vector and D a matrix of constants possibly with constraints; (3) the optimal time-consistent rule which is also linear in y; (4) ‘over-stable’ rules which have ‘too many’ stable roots. We show that rules of type (1) can be expressed and implemented as a form of integral control, all except type (2) satisfy certainty equivalence and that rules of type (4) will always be inferior to the optimal rule (1). These results are demonstrated in two illustrative examples.     

Limited-dependent rational expectations models with future expectations

This paper examines limited-dependent rational expectations (LD-RE) models containing future expectations of the dependent variable. Limited dependence is of a two-limit tobit variety which may, for example, arise as a result of a policy of imposing limits on the movement of the dependent variable by means of marginal as well as intramarginal interventions. We show that when the forcing variables are serially independent the model has an analytical solution which can be computed by backward recursion. With serially correlated forcing variables, we discuss an approximate solution method, as well as a numerically exact method that, in principle, can be implemented by stochastic simulation, although in practice it is limited by available computational capacity. The paper discusses some properties of the approximate solutions and reports the results of a limited number of Monte Carlo experiments in order to illustrate the computational feasibility of using the exact solution when the fundamentals are serially independent and the approximate solution when they are serially correlated.     

Estimation and solution of linear rational expectations models using a polynomial matrix factorization

A method to solve and estimate multivariate linear rational expectations models is described. The method is based on an iterative factorization of the polynomial matrix that describes the lags and expected leads in the model. Our experience is that the method works well in a variety of applications where other methods are either difficult or expensive to use.     

Invertible and non-invertible information sets in linear rational expectations models

Rational expectations solutions are usually derived by assuming that all state variables relevant to forward-looking behaviour are directly observable, or that they are “…an invertible function of observables” (Mehra and Prescott, 1980). Using a framework that nests linearised DSGE models, we give a number of results useful for the analysis of linear rational expectations models with restricted information sets. We distinguish between instantaneous and asymptotic invertibility, and show that the latter may require significantly less information than the former. We also show that non-invertibility of the information set can have significant implications for the time series properties of economies.     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.     

A new approach to the existence and regularity of linear equilibrium in a noisy rational expectations economy

This paper gives a new approach to show the existence and regularity of linear equilibrium established by Lou ⓡ al. (2019) for a noisy rational expectations economy. Different from the existing method which essentially requires to find a fixed point of a system of nonlinear algebraic equations, the new approach is operated directly on an alternative form of market-clearing conditions. One main advantage of the new approach is that besides homogeneous-valuation economies, it can also handle the existence of equilibrium in economies with heterogeneous valuations where the existing method for dealing with homogeneous-valuation economies fails to work.     

Determinacy,indeterminacy and dynamic misspecification in linear rational expectations models

This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model may have a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). The testing problem is addressed by a misspecification-type approach in which the overidentifying restrictions test obtained from the estimation of the system of Euler equations of the LRE model through the generalized method of moments is combined with a likelihood-based test for the cross-equation restrictions that the model places on its reduced form solution under determinacy. The resulting test has no power against a particular class of indeterminate equilibria, hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy. On the other hand, this test (i) circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy, (ii) does not involve inequality parametric restrictions and hence the use of nonstandard inference, (iii) is consistent against the dynamic misspecification of the LRE model, and (iv) is computationally simple. Monte Carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model of the US economy.     

Bootstrap of a linear model with AR-error structure

We consider a linear model with autoregressive error structure. It is shown that with probability one the distribution of the two-stage GLS estimator admits a bootstrap approximation. In a simulation study it is demonstrated that the bootstrap outperforms the normal approximation if the innovation variables are heavily correlated.     

Tests for bounded rationality with a linear dynamic model distorted by heterogeneous expectations

The purpose of this paper is to develop a methodology which will use economic data to detect the existence of boundedly rational economic agents. The bounded rationality model presented in this paper generalizes a linear dynamic rational expectations model by nesting two types of expectations. In this paper, it is claimed that the bounded rationality model as presented can be transformed into an optimal regulator problem with distortions. As a result, the methodologies developed by the optimal control theory can be used to solve the model. The likelihood function for the model is constructed by the Kalman filtering using the solution of the model. Maximum Likelihood Estimation (MLE) is performed to test for bounded rationality in the U.S. cattle market for the period from 1900 to 1990. The empirical results indicate that some fraction of economic agents in the market are boundedly rational.     

Some results on analytic optimal control solution of a simple rational expectations model

In this note, we correct a technical error in a well-known stabilization analysis in the context of a simple macroeconomic model, formulated as an optimal control problem. This error originated in a slip in an unpublished but widely circulated paper by D.A. Livesey, suggesting a procedure for obtaining a solution to a class of optimal control problems frequently encountered in a stabilization literature. We then propose an ad hoc solution method which makes clear the intuition of the solution to the particular model to which the Livesey method was applied, and of the larger class of models to which it belongs. Our solution is so formulated that the contrast between it and the Livesey result is made transparent.     

Bayesian learning and convergence to rational expectations

In this paper the probability distribution of equilibrium outcomes is assumed to be a continuous but unknown function of agents' forecasts (which are probability measures). Agents start with a prior distribution on the set of mappings from forecasts into probabilities on outcomes. This induces an initial forecast. After observing the equilibrium outcome a posterior distribution is computed which induces a new forecast. The main result is that with probability one the forecasts converge to the set of fixed points of the unknown mapping. This can be interpreted as convergence to rational expectations.     

The existence of rational expectations equilibria in a large economy with noisy price observations

Non-revealing rational expectations equilibria exist in microeconomic pure exchange economies in which a continuum of uninformed agents have suitably distributed noisy price observations. Slight dispersion in the prices observed by the subset of uninformed consumers is the key condition for continuity (and smoothness) of aggregate excess demand, although individual demands are discontinuous. It leads to equilibria in which markets approximately clear in a strong sense. The equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions.     

Estimation of the monetary model of exchange-rate determination under rational expectations

This paper develops a sticky-price version of the monetary model of exchange-rate determination under rational expectations. The reduced-form difference equation for the exchange rate has two roots which lie on either side of unity. In the solution the stable root is solved backwards and the unstable forwards. The model is estimated by the full-information maximum-likelihood method on deutsche mark-dollar data. The results are supportive of the rational-expectations restrictions.     

Identification of rational expectations models

This paper deals with the problem of the identification of simultaneous Rational Expectations (RE) models. In the case of RE models with current expectations of the endogenous variables, the necessary and sufficient conditions for the global identification are derived explicitly in terms of the structural parameters and the linear homogenous identifying restrictions. It is shown that in the absence of a priori restrictions on the processes generating the exogenous variables and the disturbances, RE models and general distributed lag models are ‘observationally equivalent’. In the case of RE models with future expectations of the endogenous variables, a general solution that highlights the ‘non-uniqueness’ problem and from which other solutions such as forward or backward solutions can be obtained, is derived. It is shown that untestable and often quite arbitrary restrictions are needed if RE models with future expectations are to be identifiable. Certain order conditions similar to those obtained for the identification of RE models with current expectations are also derived for this case.